Linear AlgebraLinear AlgebraCollege

Eigenvalues and Eigenvectors

Solve det(A - λI) = 0 and find the null space of A - λI.

An eigenvector of a square matrix $A$ is a nonzero vector that $A$ only stretches, without changing its direction: $A\mathbf{v} = \lambda\mathbf{v}$ . The scale factor $\lambda$ is the eigenvalue. Eigenvectors reveal the natural axes of a transformation, which is why they appear in vibration analysis, Google's PageRank and principal component analysis.

To find eigenvalues, solve the characteristic equation $\det(A - \lambda I) = 0$ . For each eigenvalue, solve $(A - \lambda I)\mathbf{v} = \mathbf{0}$ to get its eigenvectors. For a $2 \times 2$ matrix the eigenvalues sum to the trace and multiply to the determinant, a quick check.

Key terms

Eigenvector: A nonzero vector that A only scales.
$A\mathbf{v} = \lambda\mathbf{v}$

Key ideas

Characteristic equation: $\det(A - \lambda I) = 0$ .
Sum of eigenvalues = trace; product = determinant.
Triangular matrices have their eigenvalues on the diagonal.
A matrix with $n$ independent eigenvectors can be diagonalised: $A = PDP^{-1}$ .

Worked example

Find the eigenvalues of $A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$ .

$\det\begin{pmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{pmatrix} = (4 - \lambda)(3 - \lambda) - 2$ .
Expand: $\lambda^2 - 7\lambda + 10 = 0$ .
Factor: $(\lambda - 5)(\lambda - 2) = 0$ .
Check: $5 + 2 = 7$ = trace, $5 \times 2 = 10$ = determinant.

Answer: $\lambda = 5$ and $\lambda = 2$

Common mistakes

Subtracting $\lambda$ from every entry instead of only the diagonal.
Accepting the zero vector as an eigenvector.
Arithmetic slips in the characteristic polynomial; the trace and determinant check catches them.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Find the larger eigenvalue of $\begin{pmatrix} 3 & 0 \\ 0 & 7 \end{pmatrix}$ .
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$7$
2.The eigenvalues of a $2 \times 2$ matrix are 2 and 6. What is its determinant?
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$12$
3.Find the larger eigenvalue of $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ .
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$3$

Keep practising

Calculators

Eigenvalue Calculator